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Wildly Ramified Rigid $G_2$-Local Systems

Konstantin Jakob

Abstract

In earlier work of the author rigid irregular connections with differential Galois group $G_2$ and whose slopes have numerator $1$ were classified and new rigid connections were constructed. The same construction can be carried out for $\ell$-adic local systems in the setting of positive characteristic. In this article we provide the results that are needed to obtain the classification of wildly ramified rigid $G_2$-local systems whose slopes have numerator $1$. The overall strategy of the classification is very similar but the methods needed to obtain some invariants differ.

Wildly Ramified Rigid $G_2$-Local Systems

Abstract

In earlier work of the author rigid irregular connections with differential Galois group and whose slopes have numerator were classified and new rigid connections were constructed. The same construction can be carried out for -adic local systems in the setting of positive characteristic. In this article we provide the results that are needed to obtain the classification of wildly ramified rigid -local systems whose slopes have numerator . The overall strategy of the classification is very similar but the methods needed to obtain some invariants differ.

Paper Structure

This paper contains 4 sections, 22 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Rigid Local Systems and the Katz-Arinkin Algorithm
  3. Local Structure
  4. Classification

Key Result

Proposition 1.1

Let $\rho: I\rightarrow \textup{GL}(V)$ be an indecomposable continuous ${\overline{\mathbb{Q}}_\ell}$-representation and denote by $P$ the wild ramification subgroup of $I$. Suppose that $\rho(P^p [P,P])=1$ and that the Swan conductor $\textup{Sw}(V)<p$. In this case, $V$ is isomorphic to the repre

Theorems & Definitions (30)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1: Fu19, Thm 0.9 & Ka96, Thm 5.0.2
  • Lemma 2.2
  • proof
  • Theorem 2.3: Slope Decomposition, Ka88, 1.1.
  • Proposition 2.4: Fu15, Corollary 10.2.7
  • Theorem 2.5: Ka96, Thm 5.2.1.
  • Proposition 2.6: Ka90, Corollary 7.4.2
  • Corollary 2.7
  • ...and 20 more